L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)17-s − 0.999·20-s + (0.309 + 0.951i)25-s + 1.61·26-s + (0.5 − 0.363i)29-s − 32-s + (0.190 − 0.587i)34-s + (0.190 − 0.587i)37-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)17-s − 0.999·20-s + (0.309 + 0.951i)25-s + 1.61·26-s + (0.5 − 0.363i)29-s − 32-s + (0.190 − 0.587i)34-s + (0.190 − 0.587i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8973326186\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8973326186\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36926425756008523115270748023, −9.421290903158246070611223550822, −9.086369026612113432858997181695, −7.83603500963638250273801768274, −6.96303411695387262083330250706, −5.92677186240777542602553864875, −4.77752315961390749593433320659, −3.76785084098307981525929223850, −2.57832245448399990721496639618, −1.70109640473712372733087293475,
1.13851316143557249156083487740, 2.89091541711577560813086792368, 4.50074809890458757047069048528, 5.32952786281248233370399510258, 5.92335338374817758402859086829, 6.95117380875375078827463006553, 7.934474103368162101583956309547, 8.504052088832330572998057893537, 9.564087696617570994146296718907, 9.983600817615412636709380361596